The essential m-dissipativity for degenerate infinite dimensional stochastic Hamiltonian systems and applications

Abstract

We consider a degenerate infinite dimensional stochastic Hamiltonian system with multiplicative noise and establish the essential m-dissipativity on $L^2(\mu^{\Phi})$ of the corresponding Kolmogorov (backwards) operator. Here, $\Phi$ is the potential and $\mu^{\Phi}$ the invariant measure with density $e^{-\Phi}$ with respect to an infinite dimensional non-degenerate Gaussian measure. The main difficulty, besides the non-sectorality of the Kolmogorov operator, is the coverage of a large class of potentials. We include potentials that have neither a bounded nor a Lipschitz continuous gradient. The essential m-dissipativity is the starting point to establish the hypocoercivity of the strongly continuous contraction semigroup $(T_t)_{t\geq 0}$ generated by the Kolmogorov operator. By using the refined abstract Hilbert space hypocoercivity method of Grothaus and Stilgenbauer, originally introduced by Dolbeault, Mouhot and Schmeiser, we construct a $\mu^{\Phi}$-invariant Hunt process with weakly continuous paths and infinite lifetime, whose transition semigroup is associated with $(T_t)_{t\geq 0}$. This process provides a stochastically and analytically weak solution to the degenerate infinite dimensional stochastic Hamiltonian system with multiplicative noise. The hypocoercivity of $(T_t)_{t\geq 0}$ and the identification of $(T_t)_{t\geq 0}$ with the transition semigroup of the process leads to the exponential ergodicity. Finally, we apply our results to degenerate second order in time stochastic reaction-diffusion equations with multiplicative noise. A discussion of the class of applicable potentials and coefficients governing these equations completes our analysis.